Bounds on Gromov Hyperbolicity Constant

If $X$ is a geodesic metric space and $x_{1},x_{2},x_{3} \in X$, a geodesic triangle $T=\{x_{1},x_{2},x_{3}\}$ is the union of the three geodesics $[x_{1}x_{2}]$, $[x_{2}x_{3}]$ and $[x_{3}x_{1}]$ in $X$. The space $X$ is $\delta$-hyperbolic in the Gromov sense if any side of $T$ is contained in a $\delta$-neighborhood of the union of the two other sides, for every geodesic triangle $T$ in $X$. If $X$ is hyperbolic, we denote by $\delta(X)$ the sharp hyperbolicity constant of $X$, i.e. $\delta(X) =\inf \{ \delta\geq 0:{0.3cm}$ X ${0.2cm}$ $\text{is} {0.2cm} \delta \text{-hyperbolic} \}.$ To compute the hyperbolicity constant is a very hard problem. Then it is natural to try to bound the hyperbolycity constant in terms of some parameters of the graph. Denote by $\mathcal{G}(n,m)$ the set of graphs $G$ with $n$ vertices and $m$ edges, and such that every edge has length $1$. In this work we estimate $A(n,m):=\min\{\delta(G)\mid G \in \mathcal{G}(n,m) \}$ and $B(n,m):=\max\{\delta(G)\mid G \in \mathcal{G}(n,m) \}$. In particular, we obtain good bounds for $B(n,m)$, and we compute the precise value of $A(n,m)$ for all values of $n$ and $m$. Besides, we apply these results to random graphs.